I think you hit the nail on the head with your slow sparring example. The purpose of going slow is to be attentive. A beginner may go slow so he can remember where to put his feet. A little later, he he may know the form but be trying to feel how he balances himself. An intermedite guy may refine that to feel the connection from fingertips through C7 to the hip joints and on to the feet. An advanced guy may feel dozens of leaping off points along each circle of each joint for any number of applications.
So the outward form may remain basically the same, as may the speed of practice, but the attentive capacity of the practitioner may be many times greater for the experienced guy. When it comes time to go fast, the experienced guy doesn't have to interpret the incoming data of a punch, kick, throw, etc. because he's already felt it in practice hundreds of times. This is where taiji can follow so effectively-- it's not a matter of volitionally deciding to go where they are trying to take you, it's just a matter of relaxing and in a sense falling along any of the hundreds of paths already learned and available in the nervous system. This is a very different phenonmena than so-called "muscle memory" or any other skill learned by rote. A piano player can practice a song hundreds of times and get the notes and timing perfectly, but unless that piano player has approached the practice with beginner's mind and actually listened to the notes they were playing, then they will be lost when they try to riff with a jazz band. Of course, many people learn without consciously applying attention-- a person could just do a chin-na for example hundreds of times with hundreds of different partners and try making it work, and eventually that person would gain an intuitive sense of the adjustments they would need to make for all the slight variations in diverse situations. It has been my experience, however, that the learning is richer and more effective if we recognize what it actually is we are doing and how the learning process works, so we can become expert learners as well as expert fighters.